Algebraic Groups up to Abstract Isomorphism

With any connected affine algebraic group G over an algebraically closed field K of characteristic zero, we associate another connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abstractly isomorphic to G are precisely of the form D/α(F )×Ks +, where α is an abstract automorphism of D and s is an integer satisfying s = 0 when the derived subgroup of G contains the identity component of the center of G. It follows from the latter that any two abstractly isomorphic connected algebraic groups over K have a common algebraic central extension. The construction of D lies heavily on model theory and groups of finite Morley rank. In particular, it needs to prove that, for any two connected algebraic groups over K, the elementary equivalence of the pure groups implies they are abstractly isomorphic.